If the molecular species to be modelled is charged special
consideration must be given to obtaining a converged total energy, as
the energy of a periodically repeated system with net charge
diverges. The limit of the energy of the molecule in an infinitely
large supercell is the quantity of interest. It may be shown that this
is identical to the system containing the original charged molecule
and a uniform background charge which fills the supercell and
neutralises the charge on the molecule [9].

The energy per unit cell of this new system will converge as a power
law in the size of the supercell L. The terms of this series to
may be found and used to correct the calculated total
energy in order to obtain a more rapid convergence of the
result. These terms may be calculated in the following way.

The charge density of the system consists of the density of the
charged system and the background charge .

where if q is the charge on the molecule and V
is the volume of the supercell. This may be split into two components,
adding and subtracting a point charge at giving

The position is chosen so that has no net dipole
moment and the origin of coordinates is chosen to be at the centre of
the unit cell.

The energy of interaction may be divided into three components:

:

on a lattice interacting with itself

This is the Madelung energy of a system of point
charges on a cubic lattice immersed in a neutralising background
[10],

where is the Madelung constant dependent on the lattice.

:

on a lattice interacting with itself

It may be shown that the energy of interaction of a
neutral charged system with no net dipole moment on a cubic lattice
will converge as . [2]

:

and on a lattice interacting only
with each other.

Makov and Payne [2] show that this may
itself be divided into two components. The interaction of the point
charge with has a leading term due to the interaction of the
point charge with the second radial moment moment of , Q. This
vanishes due to symmetry for a simple cubic lattice.

The second component is due to the interaction of with the
jellium background. This leads to a total contribution of

Therefore the total calculated energy for a simple cubic lattice is
given by

where is the desired energy of the isolated charged molecule. An
example of the effect of these corrections on the convergence of the
energy of an molecule may be seen in Figure 2.3.

Figure 2.3: Graph demonstrating the effect of the
charged molecule corrections on the convergence of the energy of an
molecule with supercell side length.